# A Contrastive Divergence for Combining Variational Inference and MCMC

arXiv: https://arxiv.org/abs/1905.04062 · HTML: https://mlsanity.com/paper/1905.04062/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2019.

## Built on by (most cited first, 21 of 21)

- [MCMC Variational Inference via Uncorrected Hamiltonian Annealing](https://mlsanity.com/paper/2107.04150/) (2021)
- [Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence](https://mlsanity.com/paper/2106.15980/) (2021)
- [Data Augmentation in High Dimensional Low Sample Size Setting Using a Geometry-Based Variational Autoencoder](https://mlsanity.com/paper/2105.00026/) (2021)
- [Sliced Wasserstein Variational Inference](https://mlsanity.com/paper/2207.13177/) (2022)
- [Stein Variational Gradient Descent: many-particle and long-time asymptotics](https://mlsanity.com/paper/2102.12956/) (2021)
- [Surrogate Likelihoods for Variational Annealed Importance Sampling](https://mlsanity.com/paper/2112.12194/) (2021)
- [Learning Energy-Based Prior Model with Diffusion-Amortized MCMC](https://mlsanity.com/paper/2310.03218/) (2023)
- [Alleviating Adversarial Attacks on Variational Autoencoders with MCMC](https://mlsanity.com/paper/2203.09940/) (2022)
- [Energy based diffusion generator for efficient sampling of Boltzmann distributions](https://mlsanity.com/paper/2401.02080/) (2024)
- [Markov Chain Score Ascent: A Unifying Framework of Variational Inference with Markovian Gradients](https://mlsanity.com/paper/2206.06295/) (2022)
- [Revisiting Logistic-softmax Likelihood in Bayesian Meta-Learning for Few-Shot Classification](https://mlsanity.com/paper/2310.10379/) (2023)
- [On Divergence Measures for Training GFlowNets](https://mlsanity.com/paper/2410.09355/) (2024)
- [Optimal Regret Is Achievable with Bounded Approximate Inference Error: An Enhanced Bayesian Upper Confidence Bound Framework](https://mlsanity.com/paper/2201.12955/) (2022)
- [PAVI: Plate-Amortized Variational Inference](https://mlsanity.com/paper/2206.05111/) (2022)
- [Reparameterized Variational Rejection Sampling](https://mlsanity.com/paper/2309.14612/) (2023)
- [Differentiable Annealed Importance Sampling Minimizes The Symmetrized Kullback-Leibler Divergence Between Initial and Target Distribution](https://mlsanity.com/paper/2405.14840/) (2024)
- [PAVI: Plate-Amortized Variational Inference](https://mlsanity.com/paper/2308.16022/) (2023)
- [Bayesian Computation in Deep Learning](https://mlsanity.com/paper/2502.18300/) (2025)
- [Variational Rejection Particle Filtering](https://mlsanity.com/paper/2103.15343/) (2021)
- [Learning variational autoencoders via MCMC speed measures](https://mlsanity.com/paper/2308.13731/) (2023)
- [Amortised Inference through One-Step Implicit Sampling](https://mlsanity.com/paper/2610.12679/) (2026)

## Abstract (arXiv)

We develop a method to combine Markov chain Monte Carlo (MCMC) and variational inference (VI), leveraging the advantages of both inference approaches. Specifically, we improve the variational distribution by running a few MCMC steps. To make inference tractable, we introduce the variational contrastive divergence (VCD), a new divergence that replaces the standard Kullback-Leibler (KL) divergence used in VI. The VCD captures a notion of discrepancy between the initial variational distribution and its improved version (obtained after running the MCMC steps), and it converges asymptotically to the symmetrized KL divergence between the variational distribution and the posterior of interest. The VCD objective can be optimized efficiently with respect to the variational parameters via stochastic optimization. We show experimentally that optimizing the VCD leads to better predictive performance on two latent variable models: logistic matrix factorization and variational autoencoders (VAEs).